|\^/|     Maple V Release 4 (Worcester Polytechnic Institute)
._|\|   |/|_. Copyright (c) 1981-1996 by Waterloo Maple Inc. All rights
 \  MAPLE  /  reserved. Maple and Maple V are registered trademarks of
 <____ ____>  Waterloo Maple Inc.
      |       Type ? for help.
# January 18, 2002.   Dr. W. J. Martin
# Sample MAPLE input file.  Matrix computations.
#
# GOAL:  Solve the linear system  
#                        x1 - 2x2       = -5
#                       2x1 +  x2 +  x3 =  5
#                             3x2 - 2x3 = -4
#        using Cramer's Rule and directly.
#
> with(linalg);   # Load MAPLE's suite of linear algebra routines.
Warning, new definition for norm
Warning, new definition for trace
[BlockDiagonal, GramSchmidt, JordanBlock, LUdecomp, QRdecomp, Wronskian, addcol,
    addrow, adj, adjoint, angle, augment, backsub, band, basis, bezout,
    blockmatrix, charmat, charpoly, cholesky, col, coldim, colspace, colspan,
    companion, concat, cond, copyinto, crossprod, curl, definite, delcols,
    delrows, det, diag, diverge, dotprod, eigenvals, eigenvalues, eigenvectors,
    eigenvects, entermatrix, equal, exponential, extend, ffgausselim, fibonacci,
    forwardsub, frobenius, gausselim, gaussjord, geneqns, genmatrix, grad,
    hadamard, hermite, hessian, hilbert, htranspose, ihermite, indexfunc,
    innerprod, intbasis, inverse, ismith, issimilar, iszero, jacobian, jordan,
    kernel, laplacian, leastsqrs, linsolve, matadd, matrix, minor, minpoly,
    mulcol, mulrow, multiply, norm, normalize, nullspace, orthog, permanent,
    pivot, potential, randmatrix, randvector, rank, ratform, row, rowdim,
    rowspace, rowspan, rref, scalarmul, singularvals, smith, stack, submatrix,
    subvector, sumbasis, swapcol, swaprow, sylvester, toeplitz, trace,
    transpose, vandermonde, vecpotent, vectdim, vector, wronskian]
> 
> M := matrix
>      (3,4,    # Matrix M has 3 rows, 4 columns. Here are teh entries:
>       [ 1, -2,  0, -5,
>         2,  1,  1,  5,
>         0,  3, -2, -4]
>      );
                                [1    -2     0    -5]
                                [                   ]
                           M := [2     1     1     5]
                                [                   ]
                                [0     3    -2    -4]

> 
# Obtain the coefficient matrix:
> 
> A := submatrix( M, 1..3, 1..3);   # Select first, second, third column of M
                                   [1    -2     0]
                                   [             ]
                              A := [2     1     1]
                                   [             ]
                                   [0     3    -2]

> 
> multiply(  inverse(A),  M);
                              [1    0    0    -1]
                              [                 ]
                              [0    1    0     2]
                              [                 ]
                              [0    0    1     5]

> 
# So our solution is x = [ -1,  2,  5 ]^T
> 
# Cramer's rule. (Only works if det(A) <> 0.) 
> print(` Computing the determinant of A:`);
                         Computing the determinant of A:

> det(A);
                                      -13

> 
# In Cramer's rule, to find xi: 
#      (1)  replace the i-th column of A by the RHS 
#                      vector b = [ -5,  5,  -4 ]^T,
#      (2)  compute the determinant of this matrix,
#      (3)  divide this answer by det(A) to find xi.
> 
# Solve for x1 using Cramer's Rule:
> B1 := submatrix( M, 1..3, [4,2,3]);
                                   [-5    -2     0]
                                   [              ]
                             B1 := [ 5     1     1]
                                   [              ]
                                   [-4     3    -2]

> printf(` Determinant of B1 is %g \n`,det(B1));
 Determinant of B1 is 13 
> 
> X1 := det(B1)/det(A);
                                    X1 := -1

>        
# Solve for x2 using Cramer's Rule:
> B2 := submatrix( M, 1..3, [1,4,3]);
                                   [1    -5     0]
                                   [             ]
                             B2 := [2     5     1]
                                   [             ]
                                   [0    -4    -2]

> printf(` Determinant of B2 is %g \n`,det(B2));
 Determinant of B2 is -26 
> 
> X2 := det(B2)/det(A);
                                    X2 := 2

>        
# Solve for x3 using Cramer's Rule:
> B3 := submatrix( M, 1..3, [1,2,4]);
                                   [1    -2    -5]
                                   [             ]
                             B3 := [2     1     5]
                                   [             ]
                                   [0     3    -4]

> printf(` Determinant of B3 is %g \n`,det(B3));
 Determinant of B3 is -65 
> 
> X3 := det(B3)/det(A);
                                    X3 := 5

> quit
bytes used=1071656, alloc=917336, time=0.04